"Large $p$, small $n$" typically refers to "many independent variables, few samples".
In my case, I have $1$ independent variable, $300$ dependent variables, and $n < 20$ samples.
Thus, my case is not the typical "large $p$, small $n$" scenario (even though I do have many regression coefficients to estimate -- it's just that they are not due to the independent variables, but due to the number of dependent variables).
How is my situation classified? Does it fit into the "large $p$, small $n$" scenario, albeit I have actually less independent variables than samples?
Note: I would like to do a multivariate regression or MANOVA on my data, but have a "multicollinearity" problem. But it is not really multicollinearity because I only have one independent variable. What's going on here?